# Approximating a discrete variable with a continuous distribution - when is it appropriate?

When is it appropriate/viable to model a discrete variable with a continuous distribution? For example, say you have a class of 40 students who took two maths tests of equal difficulty on the same topic, with the tests being taken 3 weeks apart, and you record the variables $X_i = T_{1i} - T_{2i}$ where $T_{ji}$ is the jth test score of the ith child ranging from 0 to 30 (integer values only).

You would expect a histogram of your $X_i$ scores to have an outline of a rough bell curve, right? But could you use a normal distribution to model this data since the data you're working with is not continuous? Could you use a normal distribution and just use a continuity correction?

For the purposes of the question, let's say you want to know the distribution in order to do some basic eda/statistical tests.

• Normality is certainly not a requirement for your question. – SmallChess Jul 19 '17 at 16:35
• Where you say "is continuous" in the second sentence of your second paragraph, do you mean to say "is not continuous"? – Glen_b Jul 20 '17 at 3:09
• @SmallChess So generally speaking when is it appropriate to approximate with a continuous distribution when your data is discrete (as opposed to just the example I gave). Certainly in lectures we often rule out any continuous distribution to model our discrete RV, whereas for other discrete RVs we do - I'm struggling to find the consistency. Is it more to do with the assumptions of the conditions in which the random variable comes about? e.g. if you had a multiple choice quiz and a student who knew no material, a binomial distribution for the score would be more appropriate than a normal. – hhattiecc Jul 20 '17 at 7:59
• Why not you tell us what you want to do with the continuous distribution? It all depends on what you want to do. – SmallChess Jul 20 '17 at 8:01
• Of course it depends on what you're doing with it. For example it may well be that the approximation is fine if you're calculating distribution of the mean between the 2nd and 98th percentile but the approximation may be poor if you're calculating a tail quantile of the original variable. If you want general rules of thumb they'll be misleading in one case or the other. It will also depend on the actual shape of the distribution (which may in some cases be roughly bell shaped but may sometimes be rather different from that. – Glen_b Jul 20 '17 at 17:07